The Generalized Friendship Paradox for Eigenvectors

Abstract

In this paper, we investigate the generalized friendship paradox for eigenvectors (alternatively called the eigen friendship paradox and abbreviated hereafter as EFP) in the setting of inhomogeneous Erd\H{o}s--R\'enyi random graphs whose edge probabilities are generated by a continuous graphon. We consider the adjacency matrix of the graph and take the entries of the eigenvector corresponding to its largest eigenvalue as the vertex attributes. It was shown in \cite{hazra2026generalized} that the generalized friendship paradox holds in this setting. We study the empirical distribution of the resulting bias values across the vertices and derive its limiting distribution explicitly in terms of the principal eigenvalue and the corresponding eigenfunction of the integral operator whose kernel is the underlying graphon.